The Britain-based Syrian Observatory for Human Rights said at least two people were killed.
The Salafist Ahrar al-Sham movement, which calls openly for the establishment of an Islamic state after Assad's overthrow, said it carried out the bombing.
"The... Unification of the Capital brigade and the Ahrar al-Sham movement... sent a car bomb to the nesting place of the pro-regime militiamen in Mazzeh 86," a statement posted online said.
Earlier, bombs hit a police station in Rokn Eddin in north Damascus and a security branch in Bab Musalla in the southwest, killing at least eight people, according to the Observatory.
The interior ministry said 11 people died, six of them "terrorists", in the two attacks.
"Al-Nusra Front terrorists tried to enter the police station in Rokn Eddin and the criminal security branch in Damascus," it said.
"Three suicide attackers clashed with police in Rokn Eddin's police station as they prepared to detonate their explosives.
"Three other suicide bombers" tried to attack the criminal security branch in Bab Musalla, it added.
"Branch personnel fought them off and killed them, defusing their explosives," said the ministry, adding that five people besides the attackers were killed.
The ministry said investigations showed the attackers were members of Al-Nusra Front, a rebel group that has proclaimed allegiance to Al-Qaeda.
Prime Minister Wael al-Halqi condemned what he called "terrorist explosions targeting innocent citizens in Damascus" and said they showed the "desperation of the terrorists".
In northern Syria, 12 troops were killed in a rebel car bomb attack on the edges of Aleppo city, the Observatory said.
France's Hollande, on a visit to Qatar, urged the mainstream rebel Free Syrian Army to push groups like Al-Nusra out of the zones they control.
"The opposition must retake control of these areas and push these groups out," he told reporters. If extremist groups "benefit from the chaos in future, Bashar al-Assad will seize this as an excuse to continue his massacres".
Hollande later reiterated the point in Syria's southern neighbour Jordan.
"We want to help the Syrian opposition to organise themselves and clarify their relationship with extremist groups," he said.
His comments came after world powers supporting the rebels agreed on Saturday to provide them with urgent military aid.
Qatar, which hosted the gathering of foreign ministers of the "Friends of Syria", said the meeting had taken "secret decisions about practical measures to change the situation on the ground".
A final communique said "each country in its own way" would provide "urgently all the necessary material and equipment" so that the rebels could "counter brutal attacks by the regime and its allies and protect the Syrian people".
The rebels have reported receiving new equipment from "friendly" countries -- a possible allusion to Gulf Arab nations -- but the United States, France and Britain have been quiet on what they have provided.
The opposition National Coalition welcomed the increased aid to the rebels but said more was needed to bring the 27-month conflict to a conclusion.
"More steps of this decisive nature remain necessary, in order to end the conflict quickly, to stop Syrians' blood from being spilt, and to make sure their aspirations are fulfilled."
Stoking fears of a spillover of the increasingly sectarian conflict to neighbouring Lebanon, six soldiers were killed in clashes with Sunni radicals on the outskirts of the southern port city of Sidon, the army said.
Sunni cleric Sheikh Ahmad al-Assir, a fierce opponent of Hezbollah, has armed supporters who have also clashed with the powerful Shiite militant group in recent weeks.